The legal drama for Ryan Kavanaugh’s embattled Relativity Media and RKA Film Financing escalated Friday with a pair of new lawsuits filed by each party in New York state court.

RKA is doubling down on claims that Kavanaugh misappropriated funds reserved for promoting Relativity’s film slate, while Kavanaugh maintains the company is attempting to extort him and rattle other potential investors that would rescue his mini-major from collapse. RKA seeks $90 million in damages, while...